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Schrödinger equation (time-independent)
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Eigenvalue equation for stationary states.
Ĥψ = EψĤψ = Eψ, an eigenvalue equation: allowed energies are eigenvalues and the stationary states are eigenfunctions. Boundary conditions quantise energy.
For a box, ψ must vanish at the walls.
Stationary means probability density doesn't change.
In more detail
The time-independent Schrödinger equation Hψ = Eψ finds stationary states with definite energies. Solutions give the allowed energy levels and wavefunctions of bound systems.
Real-world uses
- Physicists solve it for atoms, molecules and quantum wells.
- Semiconductor engineers use it to find energy bands.
- Physicists solve the time-independent equation to find the allowed energy levels of atoms and molecules.
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Frequently asked questions
What are stationary states?
States whose probability distribution does not change with time.
Why are energies discrete?
Boundary conditions on the wavefunction restrict the allowed energy values for bound states.
What is an eigenvalue here?
The energy of a stationary state, which is the allowed value that the Hamiltonian returns when it acts on the wavefunction.
Related topics in Quantum Mechanics
Born rule · Normalisation · Schrödinger equation (time-dependent) · Operators and observables · Commutators · Particle in a box